--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Multivariate_Analysis/Caratheodory.thy Fri Aug 05 18:34:57 2016 +0200
@@ -0,0 +1,891 @@
+(* Title: HOL/Probability/Caratheodory.thy
+ Author: Lawrence C Paulson
+ Author: Johannes Hölzl, TU München
+*)
+
+section \<open>Caratheodory Extension Theorem\<close>
+
+theory Caratheodory
+ imports Measure_Space
+begin
+
+text \<open>
+ Originally from the Hurd/Coble measure theory development, translated by Lawrence Paulson.
+\<close>
+
+lemma suminf_ennreal_2dimen:
+ fixes f:: "nat \<times> nat \<Rightarrow> ennreal"
+ assumes "\<And>m. g m = (\<Sum>n. f (m,n))"
+ shows "(\<Sum>i. f (prod_decode i)) = suminf g"
+proof -
+ have g_def: "g = (\<lambda>m. (\<Sum>n. f (m,n)))"
+ using assms by (simp add: fun_eq_iff)
+ have reindex: "\<And>B. (\<Sum>x\<in>B. f (prod_decode x)) = setsum f (prod_decode ` B)"
+ by (simp add: setsum.reindex[OF inj_prod_decode] comp_def)
+ have "(SUP n. \<Sum>i<n. f (prod_decode i)) = (SUP p : UNIV \<times> UNIV. \<Sum>i<fst p. \<Sum>n<snd p. f (i, n))"
+ proof (intro SUP_eq; clarsimp simp: setsum.cartesian_product reindex)
+ fix n
+ let ?M = "\<lambda>f. Suc (Max (f ` prod_decode ` {..<n}))"
+ { fix a b x assume "x < n" and [symmetric]: "(a, b) = prod_decode x"
+ then have "a < ?M fst" "b < ?M snd"
+ by (auto intro!: Max_ge le_imp_less_Suc image_eqI) }
+ then have "setsum f (prod_decode ` {..<n}) \<le> setsum f ({..<?M fst} \<times> {..<?M snd})"
+ by (auto intro!: setsum_mono3)
+ then show "\<exists>a b. setsum f (prod_decode ` {..<n}) \<le> setsum f ({..<a} \<times> {..<b})" by auto
+ next
+ fix a b
+ let ?M = "prod_decode ` {..<Suc (Max (prod_encode ` ({..<a} \<times> {..<b})))}"
+ { fix a' b' assume "a' < a" "b' < b" then have "(a', b') \<in> ?M"
+ by (auto intro!: Max_ge le_imp_less_Suc image_eqI[where x="prod_encode (a', b')"]) }
+ then have "setsum f ({..<a} \<times> {..<b}) \<le> setsum f ?M"
+ by (auto intro!: setsum_mono3)
+ then show "\<exists>n. setsum f ({..<a} \<times> {..<b}) \<le> setsum f (prod_decode ` {..<n})"
+ by auto
+ qed
+ also have "\<dots> = (SUP p. \<Sum>i<p. \<Sum>n. f (i, n))"
+ unfolding suminf_setsum[OF summableI, symmetric]
+ by (simp add: suminf_eq_SUP SUP_pair setsum.commute[of _ "{..< fst _}"])
+ finally show ?thesis unfolding g_def
+ by (simp add: suminf_eq_SUP)
+qed
+
+subsection \<open>Characterizations of Measures\<close>
+
+definition outer_measure_space where
+ "outer_measure_space M f \<longleftrightarrow> positive M f \<and> increasing M f \<and> countably_subadditive M f"
+
+subsubsection \<open>Lambda Systems\<close>
+
+definition lambda_system :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ennreal) \<Rightarrow> 'a set set"
+where
+ "lambda_system \<Omega> M f = {l \<in> M. \<forall>x \<in> M. f (l \<inter> x) + f ((\<Omega> - l) \<inter> x) = f x}"
+
+lemma (in algebra) lambda_system_eq:
+ "lambda_system \<Omega> M f = {l \<in> M. \<forall>x \<in> M. f (x \<inter> l) + f (x - l) = f x}"
+proof -
+ have [simp]: "\<And>l x. l \<in> M \<Longrightarrow> x \<in> M \<Longrightarrow> (\<Omega> - l) \<inter> x = x - l"
+ by (metis Int_Diff Int_absorb1 Int_commute sets_into_space)
+ show ?thesis
+ by (auto simp add: lambda_system_def) (metis Int_commute)+
+qed
+
+lemma (in algebra) lambda_system_empty: "positive M f \<Longrightarrow> {} \<in> lambda_system \<Omega> M f"
+ by (auto simp add: positive_def lambda_system_eq)
+
+lemma lambda_system_sets: "x \<in> lambda_system \<Omega> M f \<Longrightarrow> x \<in> M"
+ by (simp add: lambda_system_def)
+
+lemma (in algebra) lambda_system_Compl:
+ fixes f:: "'a set \<Rightarrow> ennreal"
+ assumes x: "x \<in> lambda_system \<Omega> M f"
+ shows "\<Omega> - x \<in> lambda_system \<Omega> M f"
+proof -
+ have "x \<subseteq> \<Omega>"
+ by (metis sets_into_space lambda_system_sets x)
+ hence "\<Omega> - (\<Omega> - x) = x"
+ by (metis double_diff equalityE)
+ with x show ?thesis
+ by (force simp add: lambda_system_def ac_simps)
+qed
+
+lemma (in algebra) lambda_system_Int:
+ fixes f:: "'a set \<Rightarrow> ennreal"
+ assumes xl: "x \<in> lambda_system \<Omega> M f" and yl: "y \<in> lambda_system \<Omega> M f"
+ shows "x \<inter> y \<in> lambda_system \<Omega> M f"
+proof -
+ from xl yl show ?thesis
+ proof (auto simp add: positive_def lambda_system_eq Int)
+ fix u
+ assume x: "x \<in> M" and y: "y \<in> M" and u: "u \<in> M"
+ and fx: "\<forall>z\<in>M. f (z \<inter> x) + f (z - x) = f z"
+ and fy: "\<forall>z\<in>M. f (z \<inter> y) + f (z - y) = f z"
+ have "u - x \<inter> y \<in> M"
+ by (metis Diff Diff_Int Un u x y)
+ moreover
+ have "(u - (x \<inter> y)) \<inter> y = u \<inter> y - x" by blast
+ moreover
+ have "u - x \<inter> y - y = u - y" by blast
+ ultimately
+ have ey: "f (u - x \<inter> y) = f (u \<inter> y - x) + f (u - y)" using fy
+ by force
+ have "f (u \<inter> (x \<inter> y)) + f (u - x \<inter> y)
+ = (f (u \<inter> (x \<inter> y)) + f (u \<inter> y - x)) + f (u - y)"
+ by (simp add: ey ac_simps)
+ also have "... = (f ((u \<inter> y) \<inter> x) + f (u \<inter> y - x)) + f (u - y)"
+ by (simp add: Int_ac)
+ also have "... = f (u \<inter> y) + f (u - y)"
+ using fx [THEN bspec, of "u \<inter> y"] Int y u
+ by force
+ also have "... = f u"
+ by (metis fy u)
+ finally show "f (u \<inter> (x \<inter> y)) + f (u - x \<inter> y) = f u" .
+ qed
+qed
+
+lemma (in algebra) lambda_system_Un:
+ fixes f:: "'a set \<Rightarrow> ennreal"
+ assumes xl: "x \<in> lambda_system \<Omega> M f" and yl: "y \<in> lambda_system \<Omega> M f"
+ shows "x \<union> y \<in> lambda_system \<Omega> M f"
+proof -
+ have "(\<Omega> - x) \<inter> (\<Omega> - y) \<in> M"
+ by (metis Diff_Un Un compl_sets lambda_system_sets xl yl)
+ moreover
+ have "x \<union> y = \<Omega> - ((\<Omega> - x) \<inter> (\<Omega> - y))"
+ by auto (metis subsetD lambda_system_sets sets_into_space xl yl)+
+ ultimately show ?thesis
+ by (metis lambda_system_Compl lambda_system_Int xl yl)
+qed
+
+lemma (in algebra) lambda_system_algebra:
+ "positive M f \<Longrightarrow> algebra \<Omega> (lambda_system \<Omega> M f)"
+ apply (auto simp add: algebra_iff_Un)
+ apply (metis lambda_system_sets set_mp sets_into_space)
+ apply (metis lambda_system_empty)
+ apply (metis lambda_system_Compl)
+ apply (metis lambda_system_Un)
+ done
+
+lemma (in algebra) lambda_system_strong_additive:
+ assumes z: "z \<in> M" and disj: "x \<inter> y = {}"
+ and xl: "x \<in> lambda_system \<Omega> M f" and yl: "y \<in> lambda_system \<Omega> M f"
+ shows "f (z \<inter> (x \<union> y)) = f (z \<inter> x) + f (z \<inter> y)"
+proof -
+ have "z \<inter> x = (z \<inter> (x \<union> y)) \<inter> x" using disj by blast
+ moreover
+ have "z \<inter> y = (z \<inter> (x \<union> y)) - x" using disj by blast
+ moreover
+ have "(z \<inter> (x \<union> y)) \<in> M"
+ by (metis Int Un lambda_system_sets xl yl z)
+ ultimately show ?thesis using xl yl
+ by (simp add: lambda_system_eq)
+qed
+
+lemma (in algebra) lambda_system_additive: "additive (lambda_system \<Omega> M f) f"
+proof (auto simp add: additive_def)
+ fix x and y
+ assume disj: "x \<inter> y = {}"
+ and xl: "x \<in> lambda_system \<Omega> M f" and yl: "y \<in> lambda_system \<Omega> M f"
+ hence "x \<in> M" "y \<in> M" by (blast intro: lambda_system_sets)+
+ thus "f (x \<union> y) = f x + f y"
+ using lambda_system_strong_additive [OF top disj xl yl]
+ by (simp add: Un)
+qed
+
+lemma lambda_system_increasing: "increasing M f \<Longrightarrow> increasing (lambda_system \<Omega> M f) f"
+ by (simp add: increasing_def lambda_system_def)
+
+lemma lambda_system_positive: "positive M f \<Longrightarrow> positive (lambda_system \<Omega> M f) f"
+ by (simp add: positive_def lambda_system_def)
+
+lemma (in algebra) lambda_system_strong_sum:
+ fixes A:: "nat \<Rightarrow> 'a set" and f :: "'a set \<Rightarrow> ennreal"
+ assumes f: "positive M f" and a: "a \<in> M"
+ and A: "range A \<subseteq> lambda_system \<Omega> M f"
+ and disj: "disjoint_family A"
+ shows "(\<Sum>i = 0..<n. f (a \<inter>A i)) = f (a \<inter> (\<Union>i\<in>{0..<n}. A i))"
+proof (induct n)
+ case 0 show ?case using f by (simp add: positive_def)
+next
+ case (Suc n)
+ have 2: "A n \<inter> UNION {0..<n} A = {}" using disj
+ by (force simp add: disjoint_family_on_def neq_iff)
+ have 3: "A n \<in> lambda_system \<Omega> M f" using A
+ by blast
+ interpret l: algebra \<Omega> "lambda_system \<Omega> M f"
+ using f by (rule lambda_system_algebra)
+ have 4: "UNION {0..<n} A \<in> lambda_system \<Omega> M f"
+ using A l.UNION_in_sets by simp
+ from Suc.hyps show ?case
+ by (simp add: atLeastLessThanSuc lambda_system_strong_additive [OF a 2 3 4])
+qed
+
+lemma (in sigma_algebra) lambda_system_caratheodory:
+ assumes oms: "outer_measure_space M f"
+ and A: "range A \<subseteq> lambda_system \<Omega> M f"
+ and disj: "disjoint_family A"
+ shows "(\<Union>i. A i) \<in> lambda_system \<Omega> M f \<and> (\<Sum>i. f (A i)) = f (\<Union>i. A i)"
+proof -
+ have pos: "positive M f" and inc: "increasing M f"
+ and csa: "countably_subadditive M f"
+ by (metis oms outer_measure_space_def)+
+ have sa: "subadditive M f"
+ by (metis countably_subadditive_subadditive csa pos)
+ have A': "\<And>S. A`S \<subseteq> (lambda_system \<Omega> M f)" using A
+ by auto
+ interpret ls: algebra \<Omega> "lambda_system \<Omega> M f"
+ using pos by (rule lambda_system_algebra)
+ have A'': "range A \<subseteq> M"
+ by (metis A image_subset_iff lambda_system_sets)
+
+ have U_in: "(\<Union>i. A i) \<in> M"
+ by (metis A'' countable_UN)
+ have U_eq: "f (\<Union>i. A i) = (\<Sum>i. f (A i))"
+ proof (rule antisym)
+ show "f (\<Union>i. A i) \<le> (\<Sum>i. f (A i))"
+ using csa[unfolded countably_subadditive_def] A'' disj U_in by auto
+ have dis: "\<And>N. disjoint_family_on A {..<N}" by (intro disjoint_family_on_mono[OF _ disj]) auto
+ show "(\<Sum>i. f (A i)) \<le> f (\<Union>i. A i)"
+ using ls.additive_sum [OF lambda_system_positive[OF pos] lambda_system_additive _ A' dis] A''
+ by (intro suminf_le_const[OF summableI]) (auto intro!: increasingD[OF inc] countable_UN)
+ qed
+ have "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) = f a"
+ if a [iff]: "a \<in> M" for a
+ proof (rule antisym)
+ have "range (\<lambda>i. a \<inter> A i) \<subseteq> M" using A''
+ by blast
+ moreover
+ have "disjoint_family (\<lambda>i. a \<inter> A i)" using disj
+ by (auto simp add: disjoint_family_on_def)
+ moreover
+ have "a \<inter> (\<Union>i. A i) \<in> M"
+ by (metis Int U_in a)
+ ultimately
+ have "f (a \<inter> (\<Union>i. A i)) \<le> (\<Sum>i. f (a \<inter> A i))"
+ using csa[unfolded countably_subadditive_def, rule_format, of "(\<lambda>i. a \<inter> A i)"]
+ by (simp add: o_def)
+ hence "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) \<le> (\<Sum>i. f (a \<inter> A i)) + f (a - (\<Union>i. A i))"
+ by (rule add_right_mono)
+ also have "\<dots> \<le> f a"
+ proof (intro ennreal_suminf_bound_add)
+ fix n
+ have UNION_in: "(\<Union>i\<in>{0..<n}. A i) \<in> M"
+ by (metis A'' UNION_in_sets)
+ have le_fa: "f (UNION {0..<n} A \<inter> a) \<le> f a" using A''
+ by (blast intro: increasingD [OF inc] A'' UNION_in_sets)
+ have ls: "(\<Union>i\<in>{0..<n}. A i) \<in> lambda_system \<Omega> M f"
+ using ls.UNION_in_sets by (simp add: A)
+ hence eq_fa: "f a = f (a \<inter> (\<Union>i\<in>{0..<n}. A i)) + f (a - (\<Union>i\<in>{0..<n}. A i))"
+ by (simp add: lambda_system_eq UNION_in)
+ have "f (a - (\<Union>i. A i)) \<le> f (a - (\<Union>i\<in>{0..<n}. A i))"
+ by (blast intro: increasingD [OF inc] UNION_in U_in)
+ thus "(\<Sum>i<n. f (a \<inter> A i)) + f (a - (\<Union>i. A i)) \<le> f a"
+ by (simp add: lambda_system_strong_sum pos A disj eq_fa add_left_mono atLeast0LessThan[symmetric])
+ qed
+ finally show "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) \<le> f a"
+ by simp
+ next
+ have "f a \<le> f (a \<inter> (\<Union>i. A i) \<union> (a - (\<Union>i. A i)))"
+ by (blast intro: increasingD [OF inc] U_in)
+ also have "... \<le> f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i))"
+ by (blast intro: subadditiveD [OF sa] U_in)
+ finally show "f a \<le> f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i))" .
+ qed
+ thus ?thesis
+ by (simp add: lambda_system_eq sums_iff U_eq U_in)
+qed
+
+lemma (in sigma_algebra) caratheodory_lemma:
+ assumes oms: "outer_measure_space M f"
+ defines "L \<equiv> lambda_system \<Omega> M f"
+ shows "measure_space \<Omega> L f"
+proof -
+ have pos: "positive M f"
+ by (metis oms outer_measure_space_def)
+ have alg: "algebra \<Omega> L"
+ using lambda_system_algebra [of f, OF pos]
+ by (simp add: algebra_iff_Un L_def)
+ then
+ have "sigma_algebra \<Omega> L"
+ using lambda_system_caratheodory [OF oms]
+ by (simp add: sigma_algebra_disjoint_iff L_def)
+ moreover
+ have "countably_additive L f" "positive L f"
+ using pos lambda_system_caratheodory [OF oms]
+ by (auto simp add: lambda_system_sets L_def countably_additive_def positive_def)
+ ultimately
+ show ?thesis
+ using pos by (simp add: measure_space_def)
+qed
+
+definition outer_measure :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ennreal) \<Rightarrow> 'a set \<Rightarrow> ennreal" where
+ "outer_measure M f X =
+ (INF A:{A. range A \<subseteq> M \<and> disjoint_family A \<and> X \<subseteq> (\<Union>i. A i)}. \<Sum>i. f (A i))"
+
+lemma (in ring_of_sets) outer_measure_agrees:
+ assumes posf: "positive M f" and ca: "countably_additive M f" and s: "s \<in> M"
+ shows "outer_measure M f s = f s"
+ unfolding outer_measure_def
+proof (safe intro!: antisym INF_greatest)
+ fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" and dA: "disjoint_family A" and sA: "s \<subseteq> (\<Union>x. A x)"
+ have inc: "increasing M f"
+ by (metis additive_increasing ca countably_additive_additive posf)
+ have "f s = f (\<Union>i. A i \<inter> s)"
+ using sA by (auto simp: Int_absorb1)
+ also have "\<dots> = (\<Sum>i. f (A i \<inter> s))"
+ using sA dA A s
+ by (intro ca[unfolded countably_additive_def, rule_format, symmetric])
+ (auto simp: Int_absorb1 disjoint_family_on_def)
+ also have "... \<le> (\<Sum>i. f (A i))"
+ using A s by (auto intro!: suminf_le increasingD[OF inc])
+ finally show "f s \<le> (\<Sum>i. f (A i))" .
+next
+ have "(\<Sum>i. f (if i = 0 then s else {})) \<le> f s"
+ using positiveD1[OF posf] by (subst suminf_finite[of "{0}"]) auto
+ with s show "(INF A:{A. range A \<subseteq> M \<and> disjoint_family A \<and> s \<subseteq> UNION UNIV A}. \<Sum>i. f (A i)) \<le> f s"
+ by (intro INF_lower2[of "\<lambda>i. if i = 0 then s else {}"])
+ (auto simp: disjoint_family_on_def)
+qed
+
+lemma outer_measure_empty:
+ "positive M f \<Longrightarrow> {} \<in> M \<Longrightarrow> outer_measure M f {} = 0"
+ unfolding outer_measure_def
+ by (intro antisym INF_lower2[of "\<lambda>_. {}"]) (auto simp: disjoint_family_on_def positive_def)
+
+lemma (in ring_of_sets) positive_outer_measure:
+ assumes "positive M f" shows "positive (Pow \<Omega>) (outer_measure M f)"
+ unfolding positive_def by (auto simp: assms outer_measure_empty)
+
+lemma (in ring_of_sets) increasing_outer_measure: "increasing (Pow \<Omega>) (outer_measure M f)"
+ by (force simp: increasing_def outer_measure_def intro!: INF_greatest intro: INF_lower)
+
+lemma (in ring_of_sets) outer_measure_le:
+ assumes pos: "positive M f" and inc: "increasing M f" and A: "range A \<subseteq> M" and X: "X \<subseteq> (\<Union>i. A i)"
+ shows "outer_measure M f X \<le> (\<Sum>i. f (A i))"
+ unfolding outer_measure_def
+proof (safe intro!: INF_lower2[of "disjointed A"] del: subsetI)
+ show dA: "range (disjointed A) \<subseteq> M"
+ by (auto intro!: A range_disjointed_sets)
+ have "\<forall>n. f (disjointed A n) \<le> f (A n)"
+ by (metis increasingD [OF inc] UNIV_I dA image_subset_iff disjointed_subset A)
+ then show "(\<Sum>i. f (disjointed A i)) \<le> (\<Sum>i. f (A i))"
+ by (blast intro!: suminf_le)
+qed (auto simp: X UN_disjointed_eq disjoint_family_disjointed)
+
+lemma (in ring_of_sets) outer_measure_close:
+ "outer_measure M f X < e \<Longrightarrow> \<exists>A. range A \<subseteq> M \<and> disjoint_family A \<and> X \<subseteq> (\<Union>i. A i) \<and> (\<Sum>i. f (A i)) < e"
+ unfolding outer_measure_def INF_less_iff by auto
+
+lemma (in ring_of_sets) countably_subadditive_outer_measure:
+ assumes posf: "positive M f" and inc: "increasing M f"
+ shows "countably_subadditive (Pow \<Omega>) (outer_measure M f)"
+proof (simp add: countably_subadditive_def, safe)
+ fix A :: "nat \<Rightarrow> _" assume A: "range A \<subseteq> Pow (\<Omega>)" and sb: "(\<Union>i. A i) \<subseteq> \<Omega>"
+ let ?O = "outer_measure M f"
+ show "?O (\<Union>i. A i) \<le> (\<Sum>n. ?O (A n))"
+ proof (rule ennreal_le_epsilon)
+ fix b and e :: real assume "0 < e" "(\<Sum>n. outer_measure M f (A n)) < top"
+ then have *: "\<And>n. outer_measure M f (A n) < outer_measure M f (A n) + e * (1/2)^Suc n"
+ by (auto simp add: less_top dest!: ennreal_suminf_lessD)
+ obtain B
+ where B: "\<And>n. range (B n) \<subseteq> M"
+ and sbB: "\<And>n. A n \<subseteq> (\<Union>i. B n i)"
+ and Ble: "\<And>n. (\<Sum>i. f (B n i)) \<le> ?O (A n) + e * (1/2)^(Suc n)"
+ by (metis less_imp_le outer_measure_close[OF *])
+
+ define C where "C = case_prod B o prod_decode"
+ from B have B_in_M: "\<And>i j. B i j \<in> M"
+ by (rule range_subsetD)
+ then have C: "range C \<subseteq> M"
+ by (auto simp add: C_def split_def)
+ have A_C: "(\<Union>i. A i) \<subseteq> (\<Union>i. C i)"
+ using sbB by (auto simp add: C_def subset_eq) (metis prod.case prod_encode_inverse)
+
+ have "?O (\<Union>i. A i) \<le> ?O (\<Union>i. C i)"
+ using A_C A C by (intro increasing_outer_measure[THEN increasingD]) (auto dest!: sets_into_space)
+ also have "\<dots> \<le> (\<Sum>i. f (C i))"
+ using C by (intro outer_measure_le[OF posf inc]) auto
+ also have "\<dots> = (\<Sum>n. \<Sum>i. f (B n i))"
+ using B_in_M unfolding C_def comp_def by (intro suminf_ennreal_2dimen) auto
+ also have "\<dots> \<le> (\<Sum>n. ?O (A n) + e * (1/2) ^ Suc n)"
+ using B_in_M by (intro suminf_le suminf_nonneg allI Ble) auto
+ also have "... = (\<Sum>n. ?O (A n)) + (\<Sum>n. ennreal e * ennreal ((1/2) ^ Suc n))"
+ using \<open>0 < e\<close> by (subst suminf_add[symmetric])
+ (auto simp del: ennreal_suminf_cmult simp add: ennreal_mult[symmetric])
+ also have "\<dots> = (\<Sum>n. ?O (A n)) + e"
+ unfolding ennreal_suminf_cmult
+ by (subst suminf_ennreal_eq[OF zero_le_power power_half_series]) auto
+ finally show "?O (\<Union>i. A i) \<le> (\<Sum>n. ?O (A n)) + e" .
+ qed
+qed
+
+lemma (in ring_of_sets) outer_measure_space_outer_measure:
+ "positive M f \<Longrightarrow> increasing M f \<Longrightarrow> outer_measure_space (Pow \<Omega>) (outer_measure M f)"
+ by (simp add: outer_measure_space_def
+ positive_outer_measure increasing_outer_measure countably_subadditive_outer_measure)
+
+lemma (in ring_of_sets) algebra_subset_lambda_system:
+ assumes posf: "positive M f" and inc: "increasing M f"
+ and add: "additive M f"
+ shows "M \<subseteq> lambda_system \<Omega> (Pow \<Omega>) (outer_measure M f)"
+proof (auto dest: sets_into_space
+ simp add: algebra.lambda_system_eq [OF algebra_Pow])
+ fix x s assume x: "x \<in> M" and s: "s \<subseteq> \<Omega>"
+ have [simp]: "\<And>x. x \<in> M \<Longrightarrow> s \<inter> (\<Omega> - x) = s - x" using s
+ by blast
+ have "outer_measure M f (s \<inter> x) + outer_measure M f (s - x) \<le> outer_measure M f s"
+ unfolding outer_measure_def[of M f s]
+ proof (safe intro!: INF_greatest)
+ fix A :: "nat \<Rightarrow> 'a set" assume A: "disjoint_family A" "range A \<subseteq> M" "s \<subseteq> (\<Union>i. A i)"
+ have "outer_measure M f (s \<inter> x) \<le> (\<Sum>i. f (A i \<inter> x))"
+ unfolding outer_measure_def
+ proof (safe intro!: INF_lower2[of "\<lambda>i. A i \<inter> x"])
+ from A(1) show "disjoint_family (\<lambda>i. A i \<inter> x)"
+ by (rule disjoint_family_on_bisimulation) auto
+ qed (insert x A, auto)
+ moreover
+ have "outer_measure M f (s - x) \<le> (\<Sum>i. f (A i - x))"
+ unfolding outer_measure_def
+ proof (safe intro!: INF_lower2[of "\<lambda>i. A i - x"])
+ from A(1) show "disjoint_family (\<lambda>i. A i - x)"
+ by (rule disjoint_family_on_bisimulation) auto
+ qed (insert x A, auto)
+ ultimately have "outer_measure M f (s \<inter> x) + outer_measure M f (s - x) \<le>
+ (\<Sum>i. f (A i \<inter> x)) + (\<Sum>i. f (A i - x))" by (rule add_mono)
+ also have "\<dots> = (\<Sum>i. f (A i \<inter> x) + f (A i - x))"
+ using A(2) x posf by (subst suminf_add) (auto simp: positive_def)
+ also have "\<dots> = (\<Sum>i. f (A i))"
+ using A x
+ by (subst add[THEN additiveD, symmetric])
+ (auto intro!: arg_cong[where f=suminf] arg_cong[where f=f])
+ finally show "outer_measure M f (s \<inter> x) + outer_measure M f (s - x) \<le> (\<Sum>i. f (A i))" .
+ qed
+ moreover
+ have "outer_measure M f s \<le> outer_measure M f (s \<inter> x) + outer_measure M f (s - x)"
+ proof -
+ have "outer_measure M f s = outer_measure M f ((s \<inter> x) \<union> (s - x))"
+ by (metis Un_Diff_Int Un_commute)
+ also have "... \<le> outer_measure M f (s \<inter> x) + outer_measure M f (s - x)"
+ apply (rule subadditiveD)
+ apply (rule ring_of_sets.countably_subadditive_subadditive [OF ring_of_sets_Pow])
+ apply (simp add: positive_def outer_measure_empty[OF posf])
+ apply (rule countably_subadditive_outer_measure)
+ using s by (auto intro!: posf inc)
+ finally show ?thesis .
+ qed
+ ultimately
+ show "outer_measure M f (s \<inter> x) + outer_measure M f (s - x) = outer_measure M f s"
+ by (rule order_antisym)
+qed
+
+lemma measure_down: "measure_space \<Omega> N \<mu> \<Longrightarrow> sigma_algebra \<Omega> M \<Longrightarrow> M \<subseteq> N \<Longrightarrow> measure_space \<Omega> M \<mu>"
+ by (auto simp add: measure_space_def positive_def countably_additive_def subset_eq)
+
+subsection \<open>Caratheodory's theorem\<close>
+
+theorem (in ring_of_sets) caratheodory':
+ assumes posf: "positive M f" and ca: "countably_additive M f"
+ shows "\<exists>\<mu> :: 'a set \<Rightarrow> ennreal. (\<forall>s \<in> M. \<mu> s = f s) \<and> measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"
+proof -
+ have inc: "increasing M f"
+ by (metis additive_increasing ca countably_additive_additive posf)
+ let ?O = "outer_measure M f"
+ define ls where "ls = lambda_system \<Omega> (Pow \<Omega>) ?O"
+ have mls: "measure_space \<Omega> ls ?O"
+ using sigma_algebra.caratheodory_lemma
+ [OF sigma_algebra_Pow outer_measure_space_outer_measure [OF posf inc]]
+ by (simp add: ls_def)
+ hence sls: "sigma_algebra \<Omega> ls"
+ by (simp add: measure_space_def)
+ have "M \<subseteq> ls"
+ by (simp add: ls_def)
+ (metis ca posf inc countably_additive_additive algebra_subset_lambda_system)
+ hence sgs_sb: "sigma_sets (\<Omega>) (M) \<subseteq> ls"
+ using sigma_algebra.sigma_sets_subset [OF sls, of "M"]
+ by simp
+ have "measure_space \<Omega> (sigma_sets \<Omega> M) ?O"
+ by (rule measure_down [OF mls], rule sigma_algebra_sigma_sets)
+ (simp_all add: sgs_sb space_closed)
+ thus ?thesis using outer_measure_agrees [OF posf ca]
+ by (intro exI[of _ ?O]) auto
+qed
+
+lemma (in ring_of_sets) caratheodory_empty_continuous:
+ assumes f: "positive M f" "additive M f" and fin: "\<And>A. A \<in> M \<Longrightarrow> f A \<noteq> \<infinity>"
+ assumes cont: "\<And>A. range A \<subseteq> M \<Longrightarrow> decseq A \<Longrightarrow> (\<Inter>i. A i) = {} \<Longrightarrow> (\<lambda>i. f (A i)) \<longlonglongrightarrow> 0"
+ shows "\<exists>\<mu> :: 'a set \<Rightarrow> ennreal. (\<forall>s \<in> M. \<mu> s = f s) \<and> measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"
+proof (intro caratheodory' empty_continuous_imp_countably_additive f)
+ show "\<forall>A\<in>M. f A \<noteq> \<infinity>" using fin by auto
+qed (rule cont)
+
+subsection \<open>Volumes\<close>
+
+definition volume :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ennreal) \<Rightarrow> bool" where
+ "volume M f \<longleftrightarrow>
+ (f {} = 0) \<and> (\<forall>a\<in>M. 0 \<le> f a) \<and>
+ (\<forall>C\<subseteq>M. disjoint C \<longrightarrow> finite C \<longrightarrow> \<Union>C \<in> M \<longrightarrow> f (\<Union>C) = (\<Sum>c\<in>C. f c))"
+
+lemma volumeI:
+ assumes "f {} = 0"
+ assumes "\<And>a. a \<in> M \<Longrightarrow> 0 \<le> f a"
+ assumes "\<And>C. C \<subseteq> M \<Longrightarrow> disjoint C \<Longrightarrow> finite C \<Longrightarrow> \<Union>C \<in> M \<Longrightarrow> f (\<Union>C) = (\<Sum>c\<in>C. f c)"
+ shows "volume M f"
+ using assms by (auto simp: volume_def)
+
+lemma volume_positive:
+ "volume M f \<Longrightarrow> a \<in> M \<Longrightarrow> 0 \<le> f a"
+ by (auto simp: volume_def)
+
+lemma volume_empty:
+ "volume M f \<Longrightarrow> f {} = 0"
+ by (auto simp: volume_def)
+
+lemma volume_finite_additive:
+ assumes "volume M f"
+ assumes A: "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M" "disjoint_family_on A I" "finite I" "UNION I A \<in> M"
+ shows "f (UNION I A) = (\<Sum>i\<in>I. f (A i))"
+proof -
+ have "A`I \<subseteq> M" "disjoint (A`I)" "finite (A`I)" "\<Union>(A`I) \<in> M"
+ using A by (auto simp: disjoint_family_on_disjoint_image)
+ with \<open>volume M f\<close> have "f (\<Union>(A`I)) = (\<Sum>a\<in>A`I. f a)"
+ unfolding volume_def by blast
+ also have "\<dots> = (\<Sum>i\<in>I. f (A i))"
+ proof (subst setsum.reindex_nontrivial)
+ fix i j assume "i \<in> I" "j \<in> I" "i \<noteq> j" "A i = A j"
+ with \<open>disjoint_family_on A I\<close> have "A i = {}"
+ by (auto simp: disjoint_family_on_def)
+ then show "f (A i) = 0"
+ using volume_empty[OF \<open>volume M f\<close>] by simp
+ qed (auto intro: \<open>finite I\<close>)
+ finally show "f (UNION I A) = (\<Sum>i\<in>I. f (A i))"
+ by simp
+qed
+
+lemma (in ring_of_sets) volume_additiveI:
+ assumes pos: "\<And>a. a \<in> M \<Longrightarrow> 0 \<le> \<mu> a"
+ assumes [simp]: "\<mu> {} = 0"
+ assumes add: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<inter> b = {} \<Longrightarrow> \<mu> (a \<union> b) = \<mu> a + \<mu> b"
+ shows "volume M \<mu>"
+proof (unfold volume_def, safe)
+ fix C assume "finite C" "C \<subseteq> M" "disjoint C"
+ then show "\<mu> (\<Union>C) = setsum \<mu> C"
+ proof (induct C)
+ case (insert c C)
+ from insert(1,2,4,5) have "\<mu> (\<Union>insert c C) = \<mu> c + \<mu> (\<Union>C)"
+ by (auto intro!: add simp: disjoint_def)
+ with insert show ?case
+ by (simp add: disjoint_def)
+ qed simp
+qed fact+
+
+lemma (in semiring_of_sets) extend_volume:
+ assumes "volume M \<mu>"
+ shows "\<exists>\<mu>'. volume generated_ring \<mu>' \<and> (\<forall>a\<in>M. \<mu>' a = \<mu> a)"
+proof -
+ let ?R = generated_ring
+ have "\<forall>a\<in>?R. \<exists>m. \<exists>C\<subseteq>M. a = \<Union>C \<and> finite C \<and> disjoint C \<and> m = (\<Sum>c\<in>C. \<mu> c)"
+ by (auto simp: generated_ring_def)
+ from bchoice[OF this] guess \<mu>' .. note \<mu>'_spec = this
+
+ { fix C assume C: "C \<subseteq> M" "finite C" "disjoint C"
+ fix D assume D: "D \<subseteq> M" "finite D" "disjoint D"
+ assume "\<Union>C = \<Union>D"
+ have "(\<Sum>d\<in>D. \<mu> d) = (\<Sum>d\<in>D. \<Sum>c\<in>C. \<mu> (c \<inter> d))"
+ proof (intro setsum.cong refl)
+ fix d assume "d \<in> D"
+ have Un_eq_d: "(\<Union>c\<in>C. c \<inter> d) = d"
+ using \<open>d \<in> D\<close> \<open>\<Union>C = \<Union>D\<close> by auto
+ moreover have "\<mu> (\<Union>c\<in>C. c \<inter> d) = (\<Sum>c\<in>C. \<mu> (c \<inter> d))"
+ proof (rule volume_finite_additive)
+ { fix c assume "c \<in> C" then show "c \<inter> d \<in> M"
+ using C D \<open>d \<in> D\<close> by auto }
+ show "(\<Union>a\<in>C. a \<inter> d) \<in> M"
+ unfolding Un_eq_d using \<open>d \<in> D\<close> D by auto
+ show "disjoint_family_on (\<lambda>a. a \<inter> d) C"
+ using \<open>disjoint C\<close> by (auto simp: disjoint_family_on_def disjoint_def)
+ qed fact+
+ ultimately show "\<mu> d = (\<Sum>c\<in>C. \<mu> (c \<inter> d))" by simp
+ qed }
+ note split_sum = this
+
+ { fix C assume C: "C \<subseteq> M" "finite C" "disjoint C"
+ fix D assume D: "D \<subseteq> M" "finite D" "disjoint D"
+ assume "\<Union>C = \<Union>D"
+ with split_sum[OF C D] split_sum[OF D C]
+ have "(\<Sum>d\<in>D. \<mu> d) = (\<Sum>c\<in>C. \<mu> c)"
+ by (simp, subst setsum.commute, simp add: ac_simps) }
+ note sum_eq = this
+
+ { fix C assume C: "C \<subseteq> M" "finite C" "disjoint C"
+ then have "\<Union>C \<in> ?R" by (auto simp: generated_ring_def)
+ with \<mu>'_spec[THEN bspec, of "\<Union>C"]
+ obtain D where
+ D: "D \<subseteq> M" "finite D" "disjoint D" "\<Union>C = \<Union>D" and "\<mu>' (\<Union>C) = (\<Sum>d\<in>D. \<mu> d)"
+ by auto
+ with sum_eq[OF C D] have "\<mu>' (\<Union>C) = (\<Sum>c\<in>C. \<mu> c)" by simp }
+ note \<mu>' = this
+
+ show ?thesis
+ proof (intro exI conjI ring_of_sets.volume_additiveI[OF generating_ring] ballI)
+ fix a assume "a \<in> M" with \<mu>'[of "{a}"] show "\<mu>' a = \<mu> a"
+ by (simp add: disjoint_def)
+ next
+ fix a assume "a \<in> ?R" then guess Ca .. note Ca = this
+ with \<mu>'[of Ca] \<open>volume M \<mu>\<close>[THEN volume_positive]
+ show "0 \<le> \<mu>' a"
+ by (auto intro!: setsum_nonneg)
+ next
+ show "\<mu>' {} = 0" using \<mu>'[of "{}"] by auto
+ next
+ fix a assume "a \<in> ?R" then guess Ca .. note Ca = this
+ fix b assume "b \<in> ?R" then guess Cb .. note Cb = this
+ assume "a \<inter> b = {}"
+ with Ca Cb have "Ca \<inter> Cb \<subseteq> {{}}" by auto
+ then have C_Int_cases: "Ca \<inter> Cb = {{}} \<or> Ca \<inter> Cb = {}" by auto
+
+ from \<open>a \<inter> b = {}\<close> have "\<mu>' (\<Union>(Ca \<union> Cb)) = (\<Sum>c\<in>Ca \<union> Cb. \<mu> c)"
+ using Ca Cb by (intro \<mu>') (auto intro!: disjoint_union)
+ also have "\<dots> = (\<Sum>c\<in>Ca \<union> Cb. \<mu> c) + (\<Sum>c\<in>Ca \<inter> Cb. \<mu> c)"
+ using C_Int_cases volume_empty[OF \<open>volume M \<mu>\<close>] by (elim disjE) simp_all
+ also have "\<dots> = (\<Sum>c\<in>Ca. \<mu> c) + (\<Sum>c\<in>Cb. \<mu> c)"
+ using Ca Cb by (simp add: setsum.union_inter)
+ also have "\<dots> = \<mu>' a + \<mu>' b"
+ using Ca Cb by (simp add: \<mu>')
+ finally show "\<mu>' (a \<union> b) = \<mu>' a + \<mu>' b"
+ using Ca Cb by simp
+ qed
+qed
+
+subsubsection \<open>Caratheodory on semirings\<close>
+
+theorem (in semiring_of_sets) caratheodory:
+ assumes pos: "positive M \<mu>" and ca: "countably_additive M \<mu>"
+ shows "\<exists>\<mu>' :: 'a set \<Rightarrow> ennreal. (\<forall>s \<in> M. \<mu>' s = \<mu> s) \<and> measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>'"
+proof -
+ have "volume M \<mu>"
+ proof (rule volumeI)
+ { fix a assume "a \<in> M" then show "0 \<le> \<mu> a"
+ using pos unfolding positive_def by auto }
+ note p = this
+
+ fix C assume sets_C: "C \<subseteq> M" "\<Union>C \<in> M" and "disjoint C" "finite C"
+ have "\<exists>F'. bij_betw F' {..<card C} C"
+ by (rule finite_same_card_bij[OF _ \<open>finite C\<close>]) auto
+ then guess F' .. note F' = this
+ then have F': "C = F' ` {..< card C}" "inj_on F' {..< card C}"
+ by (auto simp: bij_betw_def)
+ { fix i j assume *: "i < card C" "j < card C" "i \<noteq> j"
+ with F' have "F' i \<in> C" "F' j \<in> C" "F' i \<noteq> F' j"
+ unfolding inj_on_def by auto
+ with \<open>disjoint C\<close>[THEN disjointD]
+ have "F' i \<inter> F' j = {}"
+ by auto }
+ note F'_disj = this
+ define F where "F i = (if i < card C then F' i else {})" for i
+ then have "disjoint_family F"
+ using F'_disj by (auto simp: disjoint_family_on_def)
+ moreover from F' have "(\<Union>i. F i) = \<Union>C"
+ by (auto simp add: F_def split: if_split_asm) blast
+ moreover have sets_F: "\<And>i. F i \<in> M"
+ using F' sets_C by (auto simp: F_def)
+ moreover note sets_C
+ ultimately have "\<mu> (\<Union>C) = (\<Sum>i. \<mu> (F i))"
+ using ca[unfolded countably_additive_def, THEN spec, of F] by auto
+ also have "\<dots> = (\<Sum>i<card C. \<mu> (F' i))"
+ proof -
+ have "(\<lambda>i. if i \<in> {..< card C} then \<mu> (F' i) else 0) sums (\<Sum>i<card C. \<mu> (F' i))"
+ by (rule sums_If_finite_set) auto
+ also have "(\<lambda>i. if i \<in> {..< card C} then \<mu> (F' i) else 0) = (\<lambda>i. \<mu> (F i))"
+ using pos by (auto simp: positive_def F_def)
+ finally show "(\<Sum>i. \<mu> (F i)) = (\<Sum>i<card C. \<mu> (F' i))"
+ by (simp add: sums_iff)
+ qed
+ also have "\<dots> = (\<Sum>c\<in>C. \<mu> c)"
+ using F'(2) by (subst (2) F') (simp add: setsum.reindex)
+ finally show "\<mu> (\<Union>C) = (\<Sum>c\<in>C. \<mu> c)" .
+ next
+ show "\<mu> {} = 0"
+ using \<open>positive M \<mu>\<close> by (rule positiveD1)
+ qed
+ from extend_volume[OF this] obtain \<mu>_r where
+ V: "volume generated_ring \<mu>_r" "\<And>a. a \<in> M \<Longrightarrow> \<mu> a = \<mu>_r a"
+ by auto
+
+ interpret G: ring_of_sets \<Omega> generated_ring
+ by (rule generating_ring)
+
+ have pos: "positive generated_ring \<mu>_r"
+ using V unfolding positive_def by (auto simp: positive_def intro!: volume_positive volume_empty)
+
+ have "countably_additive generated_ring \<mu>_r"
+ proof (rule countably_additiveI)
+ fix A' :: "nat \<Rightarrow> 'a set" assume A': "range A' \<subseteq> generated_ring" "disjoint_family A'"
+ and Un_A: "(\<Union>i. A' i) \<in> generated_ring"
+
+ from generated_ringE[OF Un_A] guess C' . note C' = this
+
+ { fix c assume "c \<in> C'"
+ moreover define A where [abs_def]: "A i = A' i \<inter> c" for i
+ ultimately have A: "range A \<subseteq> generated_ring" "disjoint_family A"
+ and Un_A: "(\<Union>i. A i) \<in> generated_ring"
+ using A' C'
+ by (auto intro!: G.Int G.finite_Union intro: generated_ringI_Basic simp: disjoint_family_on_def)
+ from A C' \<open>c \<in> C'\<close> have UN_eq: "(\<Union>i. A i) = c"
+ by (auto simp: A_def)
+
+ have "\<forall>i::nat. \<exists>f::nat \<Rightarrow> 'a set. \<mu>_r (A i) = (\<Sum>j. \<mu>_r (f j)) \<and> disjoint_family f \<and> \<Union>range f = A i \<and> (\<forall>j. f j \<in> M)"
+ (is "\<forall>i. ?P i")
+ proof
+ fix i
+ from A have Ai: "A i \<in> generated_ring" by auto
+ from generated_ringE[OF this] guess C . note C = this
+
+ have "\<exists>F'. bij_betw F' {..<card C} C"
+ by (rule finite_same_card_bij[OF _ \<open>finite C\<close>]) auto
+ then guess F .. note F = this
+ define f where [abs_def]: "f i = (if i < card C then F i else {})" for i
+ then have f: "bij_betw f {..< card C} C"
+ by (intro bij_betw_cong[THEN iffD1, OF _ F]) auto
+ with C have "\<forall>j. f j \<in> M"
+ by (auto simp: Pi_iff f_def dest!: bij_betw_imp_funcset)
+ moreover
+ from f C have d_f: "disjoint_family_on f {..<card C}"
+ by (intro disjoint_image_disjoint_family_on) (auto simp: bij_betw_def)
+ then have "disjoint_family f"
+ by (auto simp: disjoint_family_on_def f_def)
+ moreover
+ have Ai_eq: "A i = (\<Union>x<card C. f x)"
+ using f C Ai unfolding bij_betw_def by auto
+ then have "\<Union>range f = A i"
+ using f C Ai unfolding bij_betw_def
+ by (auto simp add: f_def cong del: strong_SUP_cong)
+ moreover
+ { have "(\<Sum>j. \<mu>_r (f j)) = (\<Sum>j. if j \<in> {..< card C} then \<mu>_r (f j) else 0)"
+ using volume_empty[OF V(1)] by (auto intro!: arg_cong[where f=suminf] simp: f_def)
+ also have "\<dots> = (\<Sum>j<card C. \<mu>_r (f j))"
+ by (rule sums_If_finite_set[THEN sums_unique, symmetric]) simp
+ also have "\<dots> = \<mu>_r (A i)"
+ using C f[THEN bij_betw_imp_funcset] unfolding Ai_eq
+ by (intro volume_finite_additive[OF V(1) _ d_f, symmetric])
+ (auto simp: Pi_iff Ai_eq intro: generated_ringI_Basic)
+ finally have "\<mu>_r (A i) = (\<Sum>j. \<mu>_r (f j))" .. }
+ ultimately show "?P i"
+ by blast
+ qed
+ from choice[OF this] guess f .. note f = this
+ then have UN_f_eq: "(\<Union>i. case_prod f (prod_decode i)) = (\<Union>i. A i)"
+ unfolding UN_extend_simps surj_prod_decode by (auto simp: set_eq_iff)
+
+ have d: "disjoint_family (\<lambda>i. case_prod f (prod_decode i))"
+ unfolding disjoint_family_on_def
+ proof (intro ballI impI)
+ fix m n :: nat assume "m \<noteq> n"
+ then have neq: "prod_decode m \<noteq> prod_decode n"
+ using inj_prod_decode[of UNIV] by (auto simp: inj_on_def)
+ show "case_prod f (prod_decode m) \<inter> case_prod f (prod_decode n) = {}"
+ proof cases
+ assume "fst (prod_decode m) = fst (prod_decode n)"
+ then show ?thesis
+ using neq f by (fastforce simp: disjoint_family_on_def)
+ next
+ assume neq: "fst (prod_decode m) \<noteq> fst (prod_decode n)"
+ have "case_prod f (prod_decode m) \<subseteq> A (fst (prod_decode m))"
+ "case_prod f (prod_decode n) \<subseteq> A (fst (prod_decode n))"
+ using f[THEN spec, of "fst (prod_decode m)"]
+ using f[THEN spec, of "fst (prod_decode n)"]
+ by (auto simp: set_eq_iff)
+ with f A neq show ?thesis
+ by (fastforce simp: disjoint_family_on_def subset_eq set_eq_iff)
+ qed
+ qed
+ from f have "(\<Sum>n. \<mu>_r (A n)) = (\<Sum>n. \<mu>_r (case_prod f (prod_decode n)))"
+ by (intro suminf_ennreal_2dimen[symmetric] generated_ringI_Basic)
+ (auto split: prod.split)
+ also have "\<dots> = (\<Sum>n. \<mu> (case_prod f (prod_decode n)))"
+ using f V(2) by (auto intro!: arg_cong[where f=suminf] split: prod.split)
+ also have "\<dots> = \<mu> (\<Union>i. case_prod f (prod_decode i))"
+ using f \<open>c \<in> C'\<close> C'
+ by (intro ca[unfolded countably_additive_def, rule_format])
+ (auto split: prod.split simp: UN_f_eq d UN_eq)
+ finally have "(\<Sum>n. \<mu>_r (A' n \<inter> c)) = \<mu> c"
+ using UN_f_eq UN_eq by (simp add: A_def) }
+ note eq = this
+
+ have "(\<Sum>n. \<mu>_r (A' n)) = (\<Sum>n. \<Sum>c\<in>C'. \<mu>_r (A' n \<inter> c))"
+ using C' A'
+ by (subst volume_finite_additive[symmetric, OF V(1)])
+ (auto simp: disjoint_def disjoint_family_on_def
+ intro!: G.Int G.finite_Union arg_cong[where f="\<lambda>X. suminf (\<lambda>i. \<mu>_r (X i))"] ext
+ intro: generated_ringI_Basic)
+ also have "\<dots> = (\<Sum>c\<in>C'. \<Sum>n. \<mu>_r (A' n \<inter> c))"
+ using C' A'
+ by (intro suminf_setsum G.Int G.finite_Union) (auto intro: generated_ringI_Basic)
+ also have "\<dots> = (\<Sum>c\<in>C'. \<mu>_r c)"
+ using eq V C' by (auto intro!: setsum.cong)
+ also have "\<dots> = \<mu>_r (\<Union>C')"
+ using C' Un_A
+ by (subst volume_finite_additive[symmetric, OF V(1)])
+ (auto simp: disjoint_family_on_def disjoint_def
+ intro: generated_ringI_Basic)
+ finally show "(\<Sum>n. \<mu>_r (A' n)) = \<mu>_r (\<Union>i. A' i)"
+ using C' by simp
+ qed
+ from G.caratheodory'[OF \<open>positive generated_ring \<mu>_r\<close> \<open>countably_additive generated_ring \<mu>_r\<close>]
+ guess \<mu>' ..
+ with V show ?thesis
+ unfolding sigma_sets_generated_ring_eq
+ by (intro exI[of _ \<mu>']) (auto intro: generated_ringI_Basic)
+qed
+
+lemma extend_measure_caratheodory:
+ fixes G :: "'i \<Rightarrow> 'a set"
+ assumes M: "M = extend_measure \<Omega> I G \<mu>"
+ assumes "i \<in> I"
+ assumes "semiring_of_sets \<Omega> (G ` I)"
+ assumes empty: "\<And>i. i \<in> I \<Longrightarrow> G i = {} \<Longrightarrow> \<mu> i = 0"
+ assumes inj: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> G i = G j \<Longrightarrow> \<mu> i = \<mu> j"
+ assumes nonneg: "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> \<mu> i"
+ assumes add: "\<And>A::nat \<Rightarrow> 'i. \<And>j. A \<in> UNIV \<rightarrow> I \<Longrightarrow> j \<in> I \<Longrightarrow> disjoint_family (G \<circ> A) \<Longrightarrow>
+ (\<Union>i. G (A i)) = G j \<Longrightarrow> (\<Sum>n. \<mu> (A n)) = \<mu> j"
+ shows "emeasure M (G i) = \<mu> i"
+proof -
+ interpret semiring_of_sets \<Omega> "G ` I"
+ by fact
+ have "\<forall>g\<in>G`I. \<exists>i\<in>I. g = G i"
+ by auto
+ then obtain sel where sel: "\<And>g. g \<in> G ` I \<Longrightarrow> sel g \<in> I" "\<And>g. g \<in> G ` I \<Longrightarrow> G (sel g) = g"
+ by metis
+
+ have "\<exists>\<mu>'. (\<forall>s\<in>G ` I. \<mu>' s = \<mu> (sel s)) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G ` I)) \<mu>'"
+ proof (rule caratheodory)
+ show "positive (G ` I) (\<lambda>s. \<mu> (sel s))"
+ by (auto simp: positive_def intro!: empty sel nonneg)
+ show "countably_additive (G ` I) (\<lambda>s. \<mu> (sel s))"
+ proof (rule countably_additiveI)
+ fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> G ` I" "disjoint_family A" "(\<Union>i. A i) \<in> G ` I"
+ then show "(\<Sum>i. \<mu> (sel (A i))) = \<mu> (sel (\<Union>i. A i))"
+ by (intro add) (auto simp: sel image_subset_iff_funcset comp_def Pi_iff intro!: sel)
+ qed
+ qed
+ then obtain \<mu>' where \<mu>': "\<forall>s\<in>G ` I. \<mu>' s = \<mu> (sel s)" "measure_space \<Omega> (sigma_sets \<Omega> (G ` I)) \<mu>'"
+ by metis
+
+ show ?thesis
+ proof (rule emeasure_extend_measure[OF M])
+ { fix i assume "i \<in> I" then show "\<mu>' (G i) = \<mu> i"
+ using \<mu>' by (auto intro!: inj sel) }
+ show "G ` I \<subseteq> Pow \<Omega>"
+ by fact
+ then show "positive (sets M) \<mu>'" "countably_additive (sets M) \<mu>'"
+ using \<mu>' by (simp_all add: M sets_extend_measure measure_space_def)
+ qed fact
+qed
+
+lemma extend_measure_caratheodory_pair:
+ fixes G :: "'i \<Rightarrow> 'j \<Rightarrow> 'a set"
+ assumes M: "M = extend_measure \<Omega> {(a, b). P a b} (\<lambda>(a, b). G a b) (\<lambda>(a, b). \<mu> a b)"
+ assumes "P i j"
+ assumes semiring: "semiring_of_sets \<Omega> {G a b | a b. P a b}"
+ assumes empty: "\<And>i j. P i j \<Longrightarrow> G i j = {} \<Longrightarrow> \<mu> i j = 0"
+ assumes inj: "\<And>i j k l. P i j \<Longrightarrow> P k l \<Longrightarrow> G i j = G k l \<Longrightarrow> \<mu> i j = \<mu> k l"
+ assumes nonneg: "\<And>i j. P i j \<Longrightarrow> 0 \<le> \<mu> i j"
+ assumes add: "\<And>A::nat \<Rightarrow> 'i. \<And>B::nat \<Rightarrow> 'j. \<And>j k.
+ (\<And>n. P (A n) (B n)) \<Longrightarrow> P j k \<Longrightarrow> disjoint_family (\<lambda>n. G (A n) (B n)) \<Longrightarrow>
+ (\<Union>i. G (A i) (B i)) = G j k \<Longrightarrow> (\<Sum>n. \<mu> (A n) (B n)) = \<mu> j k"
+ shows "emeasure M (G i j) = \<mu> i j"
+proof -
+ have "emeasure M ((\<lambda>(a, b). G a b) (i, j)) = (\<lambda>(a, b). \<mu> a b) (i, j)"
+ proof (rule extend_measure_caratheodory[OF M])
+ show "semiring_of_sets \<Omega> ((\<lambda>(a, b). G a b) ` {(a, b). P a b})"
+ using semiring by (simp add: image_def conj_commute)
+ next
+ fix A :: "nat \<Rightarrow> ('i \<times> 'j)" and j assume "A \<in> UNIV \<rightarrow> {(a, b). P a b}" "j \<in> {(a, b). P a b}"
+ "disjoint_family ((\<lambda>(a, b). G a b) \<circ> A)"
+ "(\<Union>i. case A i of (a, b) \<Rightarrow> G a b) = (case j of (a, b) \<Rightarrow> G a b)"
+ then show "(\<Sum>n. case A n of (a, b) \<Rightarrow> \<mu> a b) = (case j of (a, b) \<Rightarrow> \<mu> a b)"
+ using add[of "\<lambda>i. fst (A i)" "\<lambda>i. snd (A i)" "fst j" "snd j"]
+ by (simp add: split_beta' comp_def Pi_iff)
+ qed (auto split: prod.splits intro: assms)
+ then show ?thesis by simp
+qed
+
+end